The configuration spaces of parallel manipulators exhibit more nonlinearity than serial manipulators. Qualitatively, they can be seen to possess extra folds. Projection onto smaller spaces of engineering relevance, such as an output workspace or an input actuator space, these folds cast edges that exhibit boundary behavior. For example, inside the global workspace bounds of a five-bar linkage appear several local workspace bounds that only constrain certain output modes of the mechanism. The presence of such boundaries, which manifest in both input and output projections, serve as a source of confusion when these projections are studied exclusively instead of the configuration space itself. Particularly, the design of nonsymmetric parallel manipulators has been confounded by the presence of exotic projections in their input and output spaces. In this paper, we represent the configuration space with a radius graph, then weight each edge by solving an optimization problem using homotopy continuation to quantify transmission quality. We then employ a graph path planner to approximate geodesics between configuration points that avoid regions of low transmission quality. Our methodology automatically generates paths capable of transitioning between non-neighboring output modes, a motion which involves osculating multiple workspace boundaries (local, global, or both). We apply our technique to two nonsymmetric five-bar examples that demonstrate how transmission properties and other characteristics of the workspace can be selected by switching output modes.

Kinematics is the first consideration in designing the mechanical structures that comprise robots. Of the many subcategories that exist under this umbrella, an often early design goal is to achieve some desired workspace. This goal applies to both single and multi-degree-of-freedom systems. Previous literature has applied the diversity of extant optimization techniques for achieving such design goals. A conceptually simple approach to single-objective optimization is to symbolically derive a gradient vector, then find all of its zeroes. This approach is easier said than done since the resulting system is nonlinear. For this reason, sophisticated optimization heuristics are more commonly employed. In this paper, we revitalize the former approach, offering a route to efficiently find all of the gradient zeroes, including the global minimum. Our approach is facilitated by homotopy continuation. We connect the theoretical results to practical problems by demonstrating the design of a mechanism for a humanoid walking gait and the finger of a robotic hand.

Polar linkages have two degrees-of-freedom (DOF) where one input joint angle controls the length of a radial segment while another controls its angle. Considering a theoretical planar robot model, this mapping between joint angles to output motions can be shown to be energetically advantageous over the ubiquitous two-revolute linkage. Since a polar linkage’s typical construction involves a moving prismatic joint, it is cumbersome to implement alongside rotary electromagnetic actuators offsetting any advantage. In this paper, we present a procedure for designing polar linkages using only revolute joints. The procedure starts with a pre-existing single DOF straight line linkage and then finds the dimensions of a three-link attachment to produce the second DOF. In the end, the straight line linkage actuates the polar length and the attachment actuates the polar angle. The design process is framed under optimization with an objective that is both polynomial and invariant to the number of discretization points. This enables the techniques of numerical continuation to efficiently find complete sets of minima. We demonstrate our procedure with an example in which multiple minima are found including the global minimum. This computed design solution is then fabricated in order to validate the designed kinematics.